Christoffel–Darboux formula

Christoffel–Darboux formula—if a sequence of polynomials ${\displaystyle f_{0},f_{1},\dots }$ are of degrees ${\displaystyle 0,1,\dots }$, and orthogonal with respect to a probability measure ${\displaystyle \mu }$, then

${\displaystyle \sum _{j=0}^{n}{\frac {f_{j}(x)f_{j}(y)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\frac {f_{n}(y)f_{n+1}(x)-f_{n+1}(y)f_{n}(x)}{x-y}}}$

where ${\displaystyle h_{j}=\|f_{j}\|_{\mu }^{2}}$ are the squared norms, and ${\displaystyle k_{j}}$ are the leading coefficients.

Christoffel–Darboux formula, confluent form—$${\displaystyle \sum _{j=0}^{n}{\frac {f_{j}^{2}(x)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}\left[f_{n+1}'(x)f_{n}(x)-f_{n}'(x)f_{n+1}(x)\right].}$$